DSc. L'vov A.A., PhD L'vov P.A., and Matoshko I.M.

Saratov State Technical University named after Gagarin J.A., Saratov, Russia

Optimization of a multi-probe transmission line reflectometer

At present two basic techniques are used for automatic measurements of microwave circuit parameters.

The measurement technique with a multi-probe transmission line reflectometer (MTLR) is well-proved. It relies upon the analysis of distribution of a standing wave inside the line being dependent on complex reflection coefficient (CRC) of an attached load under test and absolute value of incident wave complex amplitude. Thus, it seems quite natural to use MTLR with three probes to obtain the estimates of unknown parameters. Usually, additional probes are used in order to enlarge the measurement wavelength range [1].

From one hand, the increase in the number of measuring probes allows one to exclude dramatically the influence of random noise on measurement accuracy [2] and to raise the precision of calibration as well [3]. But from the other hand, the considerable growth of probe's number may cause the appreciable distortion of standing wave pattern in the line due to own reflections of detectors. The unpredictable systematic errors being brought about by this reason shade the evident merit of MTLR: the simplicity of its construction.

Therefore, the problem is reduced to a trade-off of keeping random errors to a minimum against rising of systematic errors. Such a compromise may be achieved by use of MTLR with the limited number of measuring probes. The improvement of MTLR accuracy can be attained by means of optimization of its parameters at the development stage [4].

There is one more difficulty to be considered. Bearing in mind that optimization of MTLR parameters takes place before the particular detectors are attached to probes it is impossible to allow for the errors caused by probe (detector) gains. The authors called this source of errors 'a probe gain effect' and suggested to make up for a loss in accuracy using the effective methods of control of measurement process at the stage of ANA operation.

The paper describes parameter MTLR optimization for measurements in both narrow and wide frequency ranges and the original technique for reduction of errors due to probe gain effect by means of optimal control of measurement process.

The general MTL can be a segment of a microwave tract with regular cross-sections. Measurement probes with square-law detectors are arranged lengthwise being severe coupled with the electromagnetic field inside not to bring about the disturbances of its pattern.

The mathematical model of MTLR is well-known [1-3]:

u_j = a_j b ² {1 +r² +2r [cosj cos(4pd_j /l) + sinj sin(4pd_j /l)] } + x_j , (1)

Here u_j is a signal reading measured at the j-th probe, b is an unknown complex amplitude of standing wave in the line, a_j is the j-th probe's gain, d_j is the distance from the flange of load under test to the j-th probe, x_j is a measurement error, l is a wavelength in the MTL, r and j are modulus and phase of termination under test, N is the number of probes. The parameters d_j and l always assumed to be exactly known.

The conventional variable substitution makes this system linear with respect to new intermediate variables q_i

u_j = q₁ x_1j + q₂ x_2j + q₃ x_3j + x_j, , (2)

where

q₁ = b²(1 +r²), q₂ = b²r×cosj, q₃ = b²r×sinj, (3)

x_1j = a_j, x_2j = 2a_j×cos(4pd_j /l), x_3j = 2a_j×sin(4pd_j /l), . (4)

As a rule, the errors x_j are induced due to the shot noise of the probe detectors and thermal noise of the amplifiers. Therefore, they may be considered as an independent sample of a stationary normal random process with zero mathematical expectation and fixed variances . Thus, it is reasonable to use the maximum likelihood (ML) method [2] to obtain efficient estimators of vector = (q₁, q₂, q₃)^T (3). In this event estimation accuracy of is usually characterized by the variance matrix of estimates of these coefficients (or error matrix) [2], given by equation (5), where: y_j = 4pd_j /l are phases and are measurement weights; is an unknown coefficient; ^–1 is the designation of an inverse matrix.

, (5)

The accuracy of unknown parameters in the set (1) is entirely determined by the accuracy of intermediate parameters q₁, q₂, q₃. Furthermore, it is necessary to know the estimates of all the components to obtain the estimators of b, r, and j. In this case the theory advises to minimize the determinant of matrix D[] called the generalized variance of estimated parameters.

As it can be seen from (5) the matrix D[] depends on parameters of MTLR d_j and p_j . As it mentioned above, coefficients p_j are unknown at the development stage. Hence the optimization of MTLR can be performed only by virtue of appropriate selection of probe arrangement.

It is evident that the choice of d_j (or y_j) minimizing the determinant of D[] at the same time maximizes the one for inverse matrix D^–1[]. The problem of determinant D½D^-1[]êof matrix D^–1[] maximization is more convenient and easy.

As a trivial corollary of the Hadamard's inequality one can obtain a result that determinant D½D^–1[]ê runs into maximum if the matrix D^–1[] is diagonal, the maximum value being easily calculated

D_max = (6)

Hence the minimum value of determinant of error matrix D[] is

D_min = = (7)

where is the volume of error dispersion ellipsoid for simultaneously efficient estimator of vector . The expression (7) determines the potentially achievable accuracy of measurements by MTLR method at fixed wavelength.

If it is necessary to measure with the utmost accuracy parameters of microwave circuits in narrow wavelength range in the vicinity of central point l₀ one should minimize the generalized variance of intermediate variables (3). The probe arrangement resulted from that procedure coincides with D-optimal experiment design. In accordance with the previous discussion and expression (5) the following set of equalities has to be true

(8)

Since all p_j are unknown at the development stage it seems quite natural to assume them being equal. Then every selection of distances d_j satisfying (8) provides the optimal estimation of load under test CRC at the wavelength l₀. For example, the next equidistant probe arrangement meets the stated requirements

(9)

where N³3; k is an integer; N/2 being non-multiple of k.

The expression (9) allows a convenient arrangement of MTLR probes to be found by proper choice of parameter k. Of course there may be the other different probes' arrangements exhibiting the same quality, but they can not improve the measurement accuracy of MTLR having parameters d_j been chosen in accordance with (9).

Unfortunately, it is impossible to measure with potentially achievable accuracy at every wavelength from the wide continuous range [l_min, l_max] by means of suitable selection of MTLR probes' arrangement. Moreover, the determinant of error matrix can not reveal the optimal properties of MTLR in the range because it may be calculated only for the fixed wavelength. Hereby, it is necessary to use the optimization criterion suitable for the problem occurred.

One of the effective characterizations of MTLR accuracy may be an efficiency function (EF) of estimates [4] . It represents the ratio of error matrix determinant of parameters for current values of d_j and p_j to the one for joint efficient estimator of the same parameters at any specified wavelength l from the range [l_min, l_max]. I.e. this function is the squared fraction where numerator is dispersion ellipsoid volume for current arrangement of probes and weights and denominator is one for the optimal values of d_j, p_j at given wavelength l. It is evident that always ³1.

The weights p_j are still implied to be equal. So one should minimize the EF uniformly in the wave range [l_min, l_max] by means of appropriate selection of distances d_j from tested load flange to probes. That problem could be formulated as follows

(10)

where L_a is assured level of estimation accuracy at any arbitrary wavelength from the fixed range. The solution of (10) is rather difficult due to its nonlinearity. Therefore, only numerical techniques can provide the successful decision.

The authors developed the software for computer-aided design (CAD) of MTLR based on the minimax search of function (10). It can work in three different modes corresponding to probable formulations of optimization problem. The first statement consists in finding of the largest range (i.e. l_min and l_max) when the number of probes N is given. The basic data for the second mode are given wavelength range [l_min, l_max] and fixed N, maximum accuracy of measurements being to achieved uniformly in the whole range. Here one more relationship is added to (10)

(11)

The last mode consists in determination of minimal required number of probes to provide the measurements in the fixed range with given accuracy L_a. The decision of all the problems is performed by suitable selection of probes' arrangement. As a rule, L_a £ 1.5.

User friendly software operates in a form of a dialogue. Output data contains the distances from the load flange of MTLR to probes and the plot of versus l for the best probes' arrangement. Some effective arrangements of the limited number of MTLR probes were found. The EF plot of 8-probe TLR is shown on Fig.1. The measurement frequency (or wavelength) range (B) is near to 6 octaves.

Since the fractional change in efficiency function does not exceed 1.5 inside the required wavelength range the corresponding spread in values of the estimates of intermediate variable q₁, q₂, q₃ will not be 7% higher than that for simultaneously efficient estimator.

Fig. 1. Dependence of efficiency function on wavelength given in relative units.

But such accuracy may be insufficient if it is necessary to develop wide band ANA of high precision. Moreover, the EF depends upon weights p_j either. After the probe gains a_j are found during the calibration of MTLR the derived weights may change making it higher than L_a for some wavelengths from the range [l_min, l_max] and rising the estimation errors.

The noticed difficulties can be overcome if sequential sampling of probe detector responses is used. That way of the output data collection is rather widespread because of relatively small volume of hardware required. As it can be seen from (8) the error matrix and efficiency function depend on variances of noise errors defining a signal-to-noise ratio of probe output. It is well-known that the noise variance is reciprocally proportional to accumulation time of the detector output signal. Thus, one can vary the values of weights p_j adjusting the integration times of probes' responses. That makes possible to compensate the influence of probe gains on measurement accuracy and provide high precision of measurement at every wavelength from the range.

Actually, if all d_j and p_j are given (after calibration procedure has been performed) the relationships (8) may not be true for arbitrary l from the fixed range. But the appropriate fitting of weights p_j enables to make up for discrepancies caused by the probe gain effect. Therefore, at every measuring wavelength the redistribution of accumulation times of probe responses is to be made, the total measurement time being constant. It can be easily done using the solution of linear set (8) for unknown variables p_j, but one more equation defining the constrain for the total time of measurements should be added

(12)

Knowing relative p_j one can calculate the corresponding accumulation times to be established. From the physical point of view it is necessary to increase the integration times for 'successfully' arranged probes (closely to potential loops of the standing wave with period l) and to decrease them for 'badly' arranged probes (near the standing wave nodes).

The authors developed the software package for automatic control of the process of measurements with MTLR. It calculates the required accumulation times of probes' responses depending on the particular wavelength l and given d_j , a_j. It should be noticed that the control technique described permits one to measure the CRC of attached loads under test with practically potentially achievable accuracy at every fixed frequency from the defined range. The adjustment of weights p_j in accordance with solution of the set (8), (12) makes the error matrix D[] diagonal and hence the EF becomes equal to unity. It can be illustrated from Fig.2 where one can see the EF for the same MTLR (only small part of the whole wavelength range is shown). The upper curve shows the EF calculated for all pj = 1 and the other curves show the same EF, but with the fitted times of integration for the wavelengths l1 = 2,5 (middle plot) and l2 = 4,0 (lower plot) respectively. It can be seen from the plots that measuring accuracy becomes higher for the fixed wavelengths from the range due to appropriate control of measurements by variation of detectors' accumulation times.

The suggested technique for optimization of MTLR parameters allows the efficient estimates of load under test CRC to be obtained both in narrow and wide wavelength ranges. It was implemented by software package for computer in the form of CAD system of ANAs based on MTLR. The use of specified software in practice helped to develop the 8-probe TLR operating in the frequency range [0.5, 96] GHz (microstrip version). From the other hand the arrangement of probes in accordance with (9) enables one to measure with potentially achievable accuracy in narrow wavelength range.

The implementation of optimal control of measurement process may be useful in wide wavelength range ANAs and narrow range ones as well since it eliminates the probe gain effect. The corresponding software being developed improved dramatically the performance of 8-probe TLR in the whole frequency range mentioned above.

Finally, the use of MTLR having the limited number of optimally arranged probes (N£10) in combination with optimal calibration and data processing techniques can raise considerably the accuracy of wide frequency range ANA. Bearing in mind the cheapness and simplicity of MTLR it should be notified that manufacturing cost of wide band ANAs based on considered technique can be reduced drastically in comparison with nowadays vector ANAs without any loss in accuracy of measurements.

REFERENCES

1. R. Coldecott, "The Generalized Multiprobe Reflectometer and its Application to Automated Transmission Line Measurements", IEEE Transactions on Antennas and Propagation, vol. AP-21, no. 4, pp. 550-554, April, 1973.

2. A.A. L'vov, A.A. Mouchkaev, K.V. Semenov, “Accuracy Improvement of the Automatic Multiprobe Transmission Line Reflectometer”, The Automatic RF Techniques Group Conference Digest, ARFTG 47th, San Francisco, U.S.A., pp. 196-202.

3. A.A. L'vov and K.V. Semenov, “A Statistical Calibration Technique of the Automated Multiprobe Transmission Line Reflectometer” – Proceedings of the 10th International Conference “Systems for Automation of Engineering and Research”, September 27–29, 1996, St. Konstantin, Bulgaria, pp. 38-42.

4. B.M. Katz, A.A. L’vov, V.P. Meschanov, E.M. Shatalov, L.V. Shilova, “Synthesis of a Wideband Multiprobe Reflectometer”, IEEE Transactions on Microwave Theory and Techniques, Vol. 56, No. 2, February, 2008, P. 507-514.